$\dfrac{ -8i + 4j }{ -9 } = \dfrac{ 7i - 4k }{ 3 }$ Solve for $i$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -8i + 4j }{ -{9} } = \dfrac{ 7i - 4k }{ 3 }$ $-{9} \cdot \dfrac{ -8i + 4j }{ -{9} } = -{9} \cdot \dfrac{ 7i - 4k }{ 3 }$ $-8i + 4j = -{9} \cdot \dfrac { 7i - 4k }{ 3 }$ Reduce the right side. $-8i + 4j = -{9} \cdot \dfrac{ 7i - 4k }{ {3} }$ $-8i + 4j = -{3} \cdot \left( 7i - 4k \right)$ Distribute the right side $-8i + 4j = -{3} \cdot \left( {7i} - {4k} \right)$ $-8i + 4j = -{21}i + {12}k$ Combine $i$ terms on the left. $-{8i} + 4j = -{21i} + 12k$ ${13i} + 4j = 12k$ Move the $j$ term to the right. $13i + {4j} = 12k$ $13i = 12k - {4j}$ Isolate $i$ by dividing both sides by its coefficient. ${13}i = 12k - 4j$ $i = \dfrac{ 12k - 4j }{ {13} }$